Bilinear maps which are linear maps of products Let $V,W$ be $F$-vector spaces, where $V$ has the additional structure of (commutative) vector multiplication $* : V \times V \to V$ (for example, $V = F[X]$).
Suppose $\psi : V \to W$ is a linear map. Then $\varphi : V \times V \to W$,
$$(v_1,v_2) \mapsto \psi(v_1 * v_2)$$
is a bilinear map.
Is there a name for bilinear maps which can be described in this way?
Remark: if it was $W$ which had vector multiplication, we could define a bilinear map $(v_1,v_2) \mapsto \psi(v_1)* \psi(v_2)$.
 A: Given such a bilinear map consider the corresponding linear map $\varphi' : V \otimes V \to W$. Since, by assumption, this map factors through a map $V \otimes V \to V$, it has rank at most $\dim V$. Conversely, given any linear map $\varphi : V \otimes V \to W$ of rank at most $\dim V$, it admits some factorization through $V$. If you want the multiplication $V \otimes V \to V$ to be commutative then you run through this same argument but for symmetric bilinear maps and the symmetric square $S^2(V)$.
I'm not aware of a name for this rank condition; it doesn't seem very natural to me. If you want the multiplication to be associative then I have no idea what can be said as far as a general classification.
The special case you describe in the comments of bilinear forms on $\mathbb{R}[x]$ coming from a linear functional $L : \mathbb{R}[x] \to \mathbb{R}$ is connected to some very classical material; see umbral calculus and orthogonal polynomial for much more on this. Briefly, interesting examples of such linear functionals $L$ come from choosing a measure $\mu$ on $\mathbb{R}$ and then taking
$$L(f(x)) = \int_{\mathbb{R}} f(x) \, d \mu$$
which, if $\mu$ has infinite support, induces an inner product on $\mathbb{R}[x]$ given by
$$L(f(x) g(x)) = \int_{\mathbb{R}} f(x) g(x) \, d \mu.$$
Applying the Gram-Schmidt process to the basis $\{ 1, x, x^2, \dots \}$ of $\mathbb{R}[x]$ then gives a family of polynomials which are orthogonal with respect to the above inner product; these are the orthogonal polynomials of the measure $\mu$. This recovers many classical families of interesting polynomials that show up in various branches of mathematics and many interesting computations are possible from here. You can see, for example, my old blog post Moments, Hankel determinants, orthogonal polynomials, Motzkin paths, and continued fractions for some examples.
